Workshop on topological aspects of symplectic foliations
Université de Lyon 1 (France)
4-8 September 2017
The workshop will be devoted to the understanding of the global
topological properties of symplectic foliations in higher
dimensional manifolds. In particular, we want to study the
question if there exists a similar dichotomy for such foliations
as the one between taut/non-taut foliations in dimension 3. We
want to understand the extent to which standard symplectic
techniques apply in this new setting, under- stand how these
foliations relate to contact structures, and explore new ways to
Other perspectives and related open problems are also welcome.
The dichotomy between taut/non-taut foliations in dimension 3
extends (conjecturally) to higher dimensions by saying that a
symplectic foliation is strong if the leafwise symplectic form
arises from a global closed 2-form, and saying that it is weak
otherwise. Certain symplectic techniques only extend (naively) to
the first setting: Donaldson techniques and cohomological energy
estimates for pseudoholomorphic curves do require closeness.
The following is a tentative list of potentially interesting
topics for the workshop:
Existence of (strong) symplectic foliations. One of the main
problems of the theory is the lack of examples. The aim would
be to try to find new constructions and study whether an
h-principle can possibly hold in the weak symplectic case.
Obstructions to existence. In the opposite direction, there
are no known obstructions to the existence of strong
symplectic foliations (apart from the obvious formal ones).
Does a Novikov type theorem hold in this setting? What are
some reasonable hypothesis for such a statement?
The confoliation programme. The formal data underlying a
symplectic foliation and a contact structure is the same. Is
it possible to reproduce Eliashberg and Thurston’s
confoliation result in higher dimensions? Is it perhaps
simpler to carry it out if one assumes strongness?
Pseudoholomorphic curve theory. In the strong case, under
reasonable assumptions, moduli spaces of pseudoholomorphic
curves should be compact manifolds endowed with singular
foliations where the leaves correspond to the leafwise moduli
spaces. What structure results for strong symplectic
foliations can be obtained this way? What about classification
results for fillings of contact foliations?
Foliated Hamiltonian dynamics. What is the analogue of the
Arnold conjecture in the strong symplectic case? Is there a
meaningful foliated Lagrangian Floer theory? What about SFT?
Transverse Hamiltonian dynamics. Strong symplectic foliations
are not stable Hamiltonian usually, but there is a well
defined notion of Reeb vector field. What can be said about